Theorem 2rexrsb

Description: An equivalent expression for double restricted existence, analogous to 2exsb . (Contributed by Alexander van der Vekens, 1-Jul-2017)

	Ref	Expression
Assertion	2rexrsb	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		rexrsb	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑤 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) )
2	1	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) )
3		rexcom	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∃ 𝑤 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑤 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) )
5		rexrsb	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( 𝑥 = 𝑧 → ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ) )
6		impexp	⊢ ( ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) ↔ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑤 → 𝜑 ) ) )
7	6	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 = 𝑧 → ( 𝑦 = 𝑤 → 𝜑 ) ) )
8		r19.21v	⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 = 𝑧 → ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ( 𝑥 = 𝑧 → ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ) )
9	7 8	bitr2i	⊢ ( ( 𝑥 = 𝑧 → ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
10	9	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑥 = 𝑧 → ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
11	10	rexbii	⊢ ( ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( 𝑥 = 𝑧 → ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ) ↔ ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
12	5 11	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
13	12	rexbii	⊢ ( ∃ 𝑤 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∃ 𝑤 ∈ 𝐵 ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
14		rexcom	⊢ ( ∃ 𝑤 ∈ 𝐵 ∃ 𝑧 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
15	13 14	bitri	⊢ ( ∃ 𝑤 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑦 = 𝑤 → 𝜑 ) ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )
16	4 15	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → 𝜑 ) )